Problem: Rewrite as a simplified fraction. ${1.8\overline{65} = {?}}$
Explanation: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 1865.6565...\\ 10x &= 18.6565...\end{align*}}$ Subtracting the two equations, we have: ${990x = 1847}$ ${x = \dfrac{1847}{990}} \text{or} {1 \dfrac{857}{990}}$